Capsizing

Matlab Code for Solution of the Equation of Ship Rolling, CDF & LRFD
close all
clear all
clc
sigmahat=0.5;
significant_waveheight=0.10;
g=9.8;
A=8.1e-3*g;
B=3.11/(significant_waveheight^2);
t_start(1)=0;
t_end=900;
t_step=0.01;
number_t_step=(t_end-t_start(1))/t_step;
wave_omega_start(1)=0.05;
wave_omega_end=6;
wave_omega_step=0.05;
number_wave_omega_step=(wave_omega_end-wave_omega_start(1))/wave_omega_step;
for j=1:number_t_step+1
summation_wave_amplitude(j)=0;
end
%for i=1:number_wave_omega_step+1
%phase(i)=6.2832*rand(1,1);% Phase angle is not included in the following
for loop because
% the phase angle will be changed in each time
step
%end
load ('random_phase_for_irregularwave.mat')
for i=1:number_wave_omega_step+1 % this loop will sum the wave amplitude
at each time step over all frequencies
%S(i)=A*wave_omega_start(i)^-5*exp(-B*wave_omega_start(i)^-4);
S(i)=0.0081*((g^2)/(wave_omega_start(i)^5))*exp(-
0.032*(g/(significant_waveheight*(wave_omega_start(i)^2)))^2);
zeta(i)=sqrt(2*S(i)*wave_omega_step);
wave_amplitude=zeta(i)*cos(wave_omega_start(i)*t_start(j)+phase(i));
summation_wave_amplitude(j)=summation_wave_amplitude(j)+wave_amplitude;
if(i<number_wave_omega_step+1) %if statement is used to eleminate the
121th element of wave_omega_start
wave_omega_start(i+1)=wave_omega_start(i)+wave_omega_step;
end
end
if(j<number_t_step+1)%if statement is used to eleminate the 40002th
element of wave_omega_start
t_start(j+1)=t_start(j)+t_step;
end
end
figure
plot(t_start,summation_wave_amplitude)
title('Time vs Irregular Wave Amplitude');
xlabel('Time in second');
ylabel('Amplitude in meter');
figure
histfit(summation_wave_amplitude);

title('Histogram for Irregular Wave');
xlabel('Wave Amplitude');
ylabel('Frequency');
figure
[muhat_1,sigmahat_1] = normfit(summation_wave_amplitude);
pdfNormal_1 = normpdf(summation_wave_amplitude,muhat_1,sigmahat_1);
plot(summation_wave_amplitude, pdfNormal_1)
title('PDF for Irregular Wave');
ylabel('f(Wave Amplitude)');
figure
cdfNormal_1=normcdf(summation_wave_amplitude,muhat_1, sigmahat_1);
plot(summation_wave_amplitude,cdfNormal_1)
title('CDF for Irregular Wave');
ylabel('F(Wave Amplitude)');
time_period_roll=6.3366;
natural_omega=(2*3.1416)/time_period_roll;
density=997.04;
beam=[6.50 6.58 6.75 6.84 7.01 6.92 6.67 6.64 5.23 2.70];
draft=3.81;
displacement=227984;
metacentric_height=.947;
moment_of_inertia=(displacement*metacentric_height)/natural_omega^2;
damping_coefficient_1=0.01107;
damping_coefficient_2=0.01186;
%b_1=damping_coefficient_1/moment_of_inertia;
%b_2=damping_coefficient_2/moment_of_inertia;
b_1=0.14639;
b_2=0.14639;
for j=1:number_t_step+1% for each time step initially the summation of
heeling moment is considered 0
summation_heeling_moment(j)=0;
end
for i=1:number_wave_omega_step+1% This loop will calculte the coefficient of
right hand side/heeling moment at each frequency
time_period_wave(i)=(2*3.1416)/wave_omega_start(i);
wave_length(i)=(g/(2*3.1416))*time_period_wave(i)^2;
for j=1:number_t_step+1% this loop generates the wave amplitude at each
frequency over the defined time period.
wave_apmlitude(j)=zeta(i)*cos(wave_omega_start(i)*t_start(j)+phase(i));
end
wave_height(i)=max(wave_apmlitude)- min(wave_apmlitude);% wave height is
calculted for the wave of each frequency by subtracting minimum amplitude
from maximum amplitude
%reduction_coefficient_wave_slope(i)=atan(wave_height(i)/wave_length(i));
reduction_coefficient_wave_slope=0.682;
if (time_period_wave(i)>=11.2)
coeff_roll_add_mass=(((time_period_wave(i)-11.2)*.015)/2.7)+.07;
else
coeff_roll_add_mass=0.07-(((11.2-time_period_wave(i))*.015)/2.7);
end
%simpson rule
delata_l=1;
b44=(delata_l/3)*coeff_roll_add_mass*density*draft*beam.^3;

added_mass_moment(i)=b44(1)+b44(2)+b44(3)+b44(4)+b44(5)+b44(6)+b44(7)+b44(8)+
b44(9)+b44(10);
amplitude_e(i)=0.5*reduction_coefficient_wave_slope*(wave_height(i)/g)*wave_o
mega_start(i)^2*(abs(natural_omega^2-
(wave_omega_start(i)^2*added_mass_moment(i))/moment_of_inertia));
for j=1:number_t_step+1% this loop will sum the right hand side at each
frequency over the time period
heeling_moment=(1/(g*1000))*amplitude_e(i)*moment_of_inertia*cos(wave_omega_s
tart(i)*t_start(j)+phase(i));
summation_heeling_moment(j)=summation_heeling_moment(j)+heeling_moment;
end
end
figure
plot(t_start,summation_heeling_moment)
title('Time vs Heeling Moment');
ylabel('Heeling Moment');
figure
histfit(summation_heeling_moment);
title('Histogram for Heeling Moment');
xlabel('Heeling Moment');
figure
[muhat_2,sigmahat_2] = normfit(summation_heeling_moment);
muhat_2
sigmahat_2
pdfNormal_2 = normpdf(summation_heeling_moment,muhat_2,sigmahat_2);
plot(summation_heeling_moment, pdfNormal_2)
title('PDF for Heeling Moment');
ylabel('f(Heeling Moment)');
figure
cdfNormal_2=normcdf(summation_heeling_moment,muhat_2, sigmahat_2);
plot(summation_heeling_moment,cdfNormal_2)
title('CDF for Heeling Moment');
ylabel('F(Heeling Moment)');
f_1=inline('-b_1*v-b_2*v*abs(v)-natural_omega^2*x+summation_right_side');%
According to the runge kutta method the second order differential
%equation should be represented as :
%v_dot+b_1*v+b_2*v*abs(v)+natural_omega^2*x=summation_right_side
x_initial(1)=0;% initial roll angle is zero
v_initial(1)=0;% initial roll velocity is zero
for j=1:number_t_step+1% Right hand side is a function of time
summation_right_side_A(j)=0;%
summation_right_side_B(j)=0;
summation_right_side_C(j)=0;
end
% the following loop will sum the right hand side at first for each time
% step

for i=1:number_wave_omega_step+1% this loop wiil sum the right hand side
for each time step over all frequencies
right_hand_side_A=amplitude_e(i)*cos(wave_omega_start(i)*t_start(j)+phase(i))
;% according to the runge kutta procedure only t_start(j) is used for right
hand side
summation_right_side_A(j)=summation_right_side_A(j)+right_hand_side_A;% right
hand side is summed for each time step which will be used in dv_1.
right_hand_side_B=amplitude_e(i)*cos(wave_omega_start(i)*(t_start(j)+t_step*0
.5)+phase(i));% according to the runge kutta procedure only
t_start(j)+t_step*0.5 is used for right hand side
summation_right_side_B(j)=summation_right_side_B(j)+right_hand_side_B;%right
hand side is summed for each time step which will be used in dv_2 & dv_3.
right_hand_side_C=amplitude_e(i)*cos(wave_omega_start(i)*(t_start(j)+t_step)+
phase(i));% according to the runge kutta procedure only t_start(j)+t_step is
used for right hand side
summation_right_side_C(j)=summation_right_side_C(j)+right_hand_side_C;%right
hand side is summed for each time step which will be used in dv_4.
end
end
dx_1=t_step*v_initial(j);
dv_1=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_A(j),v_initial(j),
x_initial(j));
dx_2=t_step*(v_initial(j)+(dv_1/2));
dv_2=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_B(j),v_initial(j)+
0.5*dv_1,x_initial(j)+dx_1*0.5);
dx_3=t_step*(v_initial(j)+(dv_2/2));
dv_3=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_B(j),v_initial(j)+
0.5*dv_2,x_initial(j)+dx_2*0.5);
dx_4=t_step*(v_initial(j)+(dv_3));
dv_4=t_step*f_1(b_1,b_2,natural_omega,summation_right_side_C(j),v_initial(j)+
dv_3,x_initial(j)+dx_3);
if(j<number_t_step+1)
x_initial(j+1)=x_initial(j)+(1/6)*(dx_1+2*dx_2+2*dx_3+dx_4);%
x_initial is the roll angle
v_initial(j+1)=v_initial(j)+(1/6)*(dv_1+2*dv_2+2*dv_3+dv_4);
end
end
x_initial_1(j)=x_initial(j)*57.2975;
end
figure
plot(t_start,x_initial_1)
title('Time vs Roll Angle');
ylabel('Roll Angle in Degree');
figure

histfit(x_initial_1);
title('Histogram for Roll Angle');
xlabel('Roll Angle');
figure
[muhat_3,sigmahat_3] = normfit(x_initial_1);
pdfNormal_3 = normpdf(x_initial_1,muhat_3,sigmahat_3);
plot(x_initial_1, pdfNormal_3)
title('PDF for Roll Angle');
ylabel('f(Roll Angle)');
figure
cdfNormal_3=normcdf(x_initial_1,muhat_3, sigmahat_3);
plot(x_initial_1,cdfNormal_3)
title('CDF for Roll Angle');
ylabel('F(Roll Angle)');
righting_moment(j)=(1/1000)*displacement*metacentric_height*sin(x_initial(j))
;
end
figure
plot(t_start,righting_moment)
title('Time vs Righting Moment');
ylabel('Righting Moment');
figure
histfit(righting_moment);
title('Histogram for Righting Moment');
xlabel('Righting Moment');
figure
[muhat_4,sigmahat_4] = normfit(righting_moment);
muhat_4
sigmahat_4
pdfNormal_4 = normpdf(righting_moment,muhat_4,sigmahat_4);
plot(righting_moment, pdfNormal_4)
title('PDF for Righting Moment');
ylabel('f(Righting Moment)');
figure
cdfNormal_4=normcdf(righting_moment,muhat_4, sigmahat_4);
plot(righting_moment,cdfNormal_4)
title('CDF for Righting Moment');
ylabel('F(Righting Moment)');
%probability calculation
A=find(abs(righting_moment)<abs(summation_heeling_moment));
cases_rm_less_then_hm=length(A)
total_number_of_events=number_t_step+1
P_A=cases_rm_less_then_hm/total_number_of_events% probability that righting
moment is less then heeling moment
deck_immersion_angle=4.89;
B=find(abs(x_initial_1)>deck_immersion_angle);
cases_ra_greater_then_deack_angle=length(B)
P_B=cases_ra_greater_then_deack_angle/total_number_of_events

cases_for_A_and_B=0;
if((abs(x_initial_1(j))>deck_immersion_angle) &&
(abs(righting_moment(j))<abs(summation_heeling_moment(j))))
cases_for_A_and_B=cases_for_A_and_B+1;
end
end
cases_for_A_and_B
P_C=cases_for_A_and_B/total_number_of_events
p=1;
q=500;
r(1)=1;
for m=1:160
l=1;
for n=p:q
segment_righting_moment_(m,l)=abs(righting_moment(n));
segment_heeling_moment_(m,l)=abs(summation_heeling_moment(n));
l=l+1;
end
[segment_muhat_4(m),segment_sigmahat_4(m)]=normfit(segment_righting_moment_(m
,1:500));
[segment_muhat_2(m),segment_sigmahat_2(m)]=normfit(segment_heeling_moment_(m,
1:500));
z(m)=(segment_muhat_4(m)-
segment_muhat_2(m))/sqrt(segment_sigmahat_4(m)*segment_sigmahat_4(m)+segment_
sigmahat_4(m)*segment_sigmahat_4(m));
p=p+500;
q=q+500;
if(m<160)%if statement is used to eleminate the 40002th element of
wave_omega_start
r(m+1)=r(m)+1;
end
end
figure
plot(r,z)
title('"Z" values for the each segment from the total 800 second window');
xlabel('Segment No');
ylabel('"Z"');
p_normal=normcdf(z);
figure
plot(r,p_normal)
title('probability for the each segment from the total 800 second window');
xlabel('Segment No');
ylabel('probability');
p=1;
q=40;
r1(1)=1;
for m=1:4
average_p_normal(m)=0;
for n=p:q
average_p_normal(m)=average_p_normal(m)+p_normal(n);
end

mean_p_normal(m)=average_p_normal(m)/40;
p=p+40;
q=q+40;
if(m<4)%if statement is used to eleminate the 40002th element of
wave_omega_start
r1(m+1)=r1(m)+1;
end
end
pp = polyfit(r1,mean_p_normal,3);
x1 = linspace(1,4);
y1_2 = polyval(pp,x1);
figure
plot(r1,mean_p_normal,'o')
hold on
plot(x1,y1_2)
hold off
save('average_p_normal.mat','y1_2','-append')
Matlab Code for the Comparison of Capsizing Probability for Different Sea State
close all
clear all
clc
load('average_p_normal.mat')
figure
plot(x1,y1_1,'k')
hold on
plot(x1,y1_2,'c')
hold on
plot(x1,y1_3,'b')
hold on
plot(x1,y1_4,'m')
hold on
plot(x1,y1_5,'g')
hold on
plot(x1,y1_6,'y')
hold on
plot(x1,y1_7,'r')
hold on
ylabel('Probaility of Capsizing')
xlabel('segment no')
legend('Sea Stae 1','Sea state 2','Sea state 3','Sea state 4','Sea state
5','Sea state 6','Sea state 7')
box off
a2 = axes('XAxisLocation', 'Top');
set(a2, 'color', 'none')
set(a2, 'YTick', [])
set(a2, 'XLim', [0 800])
xlabel(a2,'Time')

Capsizing

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